Semitopological group explained

In mathematics, a semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group; all topological groups are semitopological groups but the converse does not hold.

Formal definition

A semitopological group

G is a topological space that is also a group such thatg _{1:} G x G \to G : (x,y)\mapsto xy

is continuous with respect to both

x and y . (Note that a topological group is continuous with reference to both variables simultaneously, and g _{2:} G\to G : x \mapsto x ^{-1} is also required to be continuous. Here G x G is viewed as a topological space with the product topology.)^[1] ( R , + ) with its usual structure as an additive abelian group. Apply the lower limit topology to R with topological basis the family \{[a,b): -inf ty < a < b < inf ty \} . Then g _{1} is continuous, but g _{2} is not continuous at 0: [ 0 ,b) is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in g ^{-1} _{2} ([ 0 ,b)) .

It is known that any locally compact Hausdorff semitopological group is a topological group.^[2] Other similar results are also known.^[3]

See also

• Lie group
• Algebraic group
• Compact group
• Topological ring

Notes and References

1. Book: Husain. Taqdir. Introduction to Topological Groups. 2018. Courier Dover Publications. 9780486828206. 27. en.
2. Book: Arhangel’skii. Alexander. Tkachenko. Mikhail. Topological Groups and Related Structures, An Introduction to Topological Algebra.. 2008. Springer Science & Business Media. 9789491216350. 114. en.
3. Book: Aull. C. E.. Lowen. R.. Handbook of the History of General Topology. 2013. Springer Science & Business Media. 9789401704700. 1119. en.